FET - Open Domain IST-2001-35511FET - Open Domain IST-2001-35511

DALHM Development and Analysis of Left Handed

Materials

FORTH, Crete, GreeceBilkent University, Ankara, TurkeyImperial College, London, England

2nd year MeetingJuly 29-30, 2004Crete, Greece

Computational Methods Plane wave expansion method (PWE) R. Moussa, S. Foteinopoulou & M. Kafesaki

Transfer matrix method (TMM) Th. Koschny, R. Penciu & P. Markos

Finite-difference-time-domain-method (FDTD) M. Kafesaki, R. Moussa, & S. Foteinopoulou

Effective medium theories E. N. Economou, Th. Koschny

Microwave studio T.O.Gundogdu, R. Penciu, M. Kafesaki & Lei

Zhang

Transfer matrix method to compute scattering amplitudes

New discretization scheme: Symmetry is preserved

This new symmetric material discretization completely eliminates the problem of the off-diagonal terms in the transfer matrix approach for sufficiently accurate computation. So we have successfully implemented a new discretization scheme that gives no off-diagonal terms.

continuum Homogeneous Effective Medium inversion

Generic LH related Metamaterials

€

′ ω p

€

′ ω p

€

ωm

€

ωm

€

′ ω m

€

′ ω m

€

ωa/c

€

ωa/c

€

ωa/c

€

ε

€

μ

Resonance and anti-resonance

Typical LHM behavior

Analytic model for the electric and magnetic response of SRRs

Analytic model of the electric and magnetic response of LHMs

PRL (accepted, 2004)

Electric response of LHM

Electric and magnetic response of SRR

E and M response of LHM

Electric response of wires

Electric response of cut wires

f (GHz)

T

30 GHz FORTH structure with 600 x 500 x 500 μm3

SubstrateGaAsεb=12.3

LHM Design used by UCSD, Bilkent and ISU

LHMSRRClosed LHM

Left-Handed MaterialsLeft-Handed Materials

SRR Parameters:

r1=2.5 mm, r2=3.6 mm,d=w=0.2 mmt=0.9 mm

Parameters:

ax=9.3 mm ay=9mm az=6.5 mmNx=15 Ny=15

w

t

dr1

r2

Transmission data for open and closed SRRs

Bilkent & Forth

Magnetic resonancedisappears for closed SRRs

Bilkent & Forth

Effective ωp of closed SRRs & wires is much lower than ωp of the wires.

Best LH peak in a left-handed material

Losses: -0.3 dB/cm Bilkent & Forth

Peak at f=4 GHz=75 mm

much largerthan

size of SRR

a=3.6 mm

Experiment Theory

Bilkent & Forth

Bilkent & Forth

Retrieval parameters for Bilkent structure

Transmission spectra in the low frequency region for 3 unit cells

Transmission spectra in the higher frequency region

Transmission S21 in the lower and higher region (1 unit cell)

10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0

0.01

0.1

1

S21

Frequency(GHz)2.0 2.5 3.0 3.5 4.0

0.01

0.1

1

S21

Frequency(GHz)

Retrieved n in the lower frequency region

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0-6

-5

-4

-3

-2

-1

0

1

Frequency(GHz)

real n imag n

n

10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5-10

-5

0

5

10

n

Frequency(GHz)

real n imag n

Retrieved n in the higher frequency region

Retrieved ε, μ in the lower frequency region

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0-30

-25

-20

-15

-10

-5

0

5

10

Frequency(GHz)

ε μ

ε, μ

Retrieved ε, μ in the higher region

10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5-10

-5

0

5

10

15

ε, μ

( )Frequency GHz

ε μ

Closed rings

2 4 6 8 10 12-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

n real n imag n

Frequency(GHz)

Closed rings

Closed rings

2 4 6 8 10 12-15

-10

-5

0

5

10

15

ε μ

ε, μ

( )Frequency GHz

Electric and Magnetic Response of SRRs and LHMs

• Electric and Magnetic Response are independent.

• One can change the magnetic response without changing the electric response.

• GHz and THz magnetic response in artificial structures!

• The SRR has strong electric response. It’s cut-wire like.

• Effective electric response of LHM is the sum of wire and SRR.

• Effective ωp of the LHM is much lower than ωp of the wires.

• There are “phony” LH peaks when ωp < ωm PRL (accepted, 2004)

Electric coupling to the magnetic resonance

APL 84, 2943 (2004)

8 9 10 11-10

0

10

20

30 retrieved μ ( )for a

realμ imagμμ

( )Frequency GHz

8 9 10 11-10

0

10

20

30 retrieved ε ( )for d

ε

realε imagε

( )Frequency GHz8 9 10 11

0

1

2

3

4 retrieved μ ( )for d

μ

( )Frequency GHz

realμ imagμ

Photonics and Nanostructures (accepted, 2004)

Magnetic response at 100 THz, almost optical frequencies

S. Linden & M. Wegener, Karlsruhe 10

Magnetic response at 100 THz, almost optical frequencies

S. Linden & M. Wegener, Karlsruhe

Magnetic response at 100 THz, almost optical frequencies

S. Linden & M. Wegener, Karlsruhe

4 cases of different propagation and polarization for single ring

cell = 2.5mm

gap azimuthal = 0.3mm

ring outer side length = 2.2mm

ring width = 0.2mm

sub thickness =0.25mm

Transmission and retrieved parameters

4 6 8 10 12 14 16 18 20-6

-4

-2

0

2

4

6

Frequency(GHz)

kparEpar kparEpen kpenEpar kpenEpen

μ

4 6 8 10 12 14 16 18 20-20

-10

0

10

20

30

40

50

60

Frequency(GHz)

kparEpar kparEpen kpenEpar kpenEpenε

4 6 8 10 12 14 16 18 200.0

0.2

0.4

0.6

0.8

1.0

kparEpar kparEpen kpenEpar kpenEpen

S21

Frequency(GHz)

k in the plane gives a negative μ region,

otherwise μ remains positive even though

a gap appears in the transmission spectra

when E field is along the ring gap.

Opposed ring can get rid of the effect of electric coupling

2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

kparEpar kparEpen kpenEpar kpenEpen

S21

frequency(GHz)

2 4 6 8 10 12 14-4

-2

0

2

4

6

kparEpar kparEpen kpenEpar kpenEpen

frequency(GHz)

μ

2 4 6 8 10 12 14-4

0

4

8

12

16

20

24

28

32

kparEpar kparEpen kpenEpar kpenEpenε

( )frequency GHz

Sub thickness dependence

the closer the separated opposed rings are, the weaker the electric coupling is.Here are shown transmission spectra and μ when the thickness are chosen to be

0.25mm, 0.125mm and 0.075mm

2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

t = 0.25 t = 0.25 * 0.5 t = 0.25 * 0.3

S21

frequency (GHz)2 4 6 8 10 12 14

0.6

0.8

1.0

1.2

1.4

t = 0.25mmt = 0.25mm*0.5t = 0.25mm*0.3

μ

frequency

3D Rings & Wirescell = 2.5mm in X/Y/Zring side length = 2.2mmwire width = 0.2mmring width = 0.2mmopposed ring separation = 0.2mm

this structure is symmetric in 3D and also behaves almost the same for different polarizationssee black and red curve below.

6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

kxEz kxEy

S21

Frequency (GHz)

Retrieved Z, n

9 10 11 12 13 14 150

2

4

6

8 kxEz

Frequency (GHz)

Z

9 10 11 12 13 14 15-6

-5

-4

-3

-2

-1

0

Frequency (GHz)

n

n and ε μ

6 8 10 12 14-50

-40

-30

-20

-10

0

10

20

Frequency(GHz)

real ε realμ

ε μ

there are multiple negative index regions from the retrieval code

6 8 10 12 14-10

-8

-6

-4

-2

0

real n imag n

n

Frequency(GHz)

Going to multi-gap structures (1)

(a) better than (b) (wider SRR dip); (c) better than (d) (stronger dip); (e) like the conventional SRR but weaker dip (for large separation)

Problem: Increase of ωm (ωm close to ω0 )

Gaps act like capacitors in series: ωm2(n gaps)

~ n ωm2(1 gap)

Reason: requirement for higher symmetry, for use in 3D LH structures

a) b) c) d) e)

Going to multi-gap structures (2)

Solution: Make the gaps smaller or change the design

Improvements?

Up to a point

Only the left one

Promising multi-gap structures from 1D study

a) b)

(a): Detailed study on progress (in 1D)

(b): Not studied in detail yet

(c): Good LH T

3D structures

a) b) c) Best combination: (b)+(c)

c)

Two-sided SRR Structures: No coupling to Electric Field

2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

t = 0.25 t = 0.25 * 0.5 t = 0.25 * 0.3

S21

frequency (GHz)

2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

kparEpar kparEpen kpenEpar kpenEpen

S21

frequency(GHz)

Two-sided SRRs do not have coupling to electric field

Multi-Azimuthal SRR

2-gaps SRR

• adding another gap at the opposite side of ring helps to inhibit the electric coupling

• Magnetic properties remain the same

testing with different k and polarization

16 18 20 22 24 26 28 30-1

0

1

2

3

Frequency(GHz)

retrieved shows that there is no magneticresonance when K is out of the SRR plane

K||e|_ K||e|| K|_e|_ K|_e||μ

4-gaps SRR

introducing 4 cuts at each side of the ring respectively may help to build a near-isotropic structure

unit cell = 25mm

ring side = 22mm

ring width = 2mm

azimuthal = 50um

sub thickness = 2.5mm

20 25 30 35 40 45 50-0.5

0.0

0.5

1.0

1.5

2.0

realmu immu

Frequency (GHz)

μ

20 25 30 35 40 45 50-1.0

-0.5

0.0

0.5

1.0

ren imn

n

Frequency (GHz)20 25 30 35 40 45 50

-4

-3

-2

-1

0

1

2

3

4

Frequency (GHz)

rez imz

Z

Comparison of 1/2/4-gaps SRRall SRR have a total azimuthal length 50μm

While keeping the sum of the gap widths the same, the 1, 2 or 4 cuts SRR have different resonance frequencies. 4-cuts SRR has a much higher resonance region than single cut ones. To build a LHM with 4cuts SRR, wires need to be more compact to enhance plasma frequency.

LHM composed of 4-cuts SRR and Wires

• Unit cell dimension = 2.5mm

• Ring side length = 2.2mm

• Ring width = 0.2mm

• Wire width = 0.6mm

• Ring thickness = 17micron

• Wire thickness = 50micron

• Sub thickness = 0.25mm

• Sub permittivity = 2.2

• Deposit permittivity = 9.61

18 20 22 24 26 28-6

-5

-4

-3

-2

-1

0

1

retrivied n

Frequency

real(n) imag(n)

n

18 20 22 24 26 28-6

-4

-2

0

2

4

6

8retrivied ε, μ

Frequency

(realε) (realμ)

ε,μ

T and R of a Metamaterial

€

ts =exp(−ikd)

cos nkd( )−1

2z +

1

z ⎛ ⎝

⎞ ⎠sin nkd( )

€

z =με

€

n = με

€

μ ω( )=1−ωmp

2

ω 2 −ω m02 + iΓm0

€

ε ω( ) =1−ωep

2

ω2 −ω e02 + iΓe 0

UCSD and ISU, PRB, 65, 195103 (2002)

€

rs = − ts exp(+ikd)i(z −1 / z)sin(nkd) / 2

d

z, n

Inversion of S-parameters

d

UCSD and ISU, PRB, 65, 195103 (2002)

€

e ik

€

teik

€

re− ik

€

ε =nz

€

μ =nz

€

n =1kd

cos−1 12 ′ t

1− r 2 − ′ t 2( )[ ]

⎛ ⎝

⎞ ⎠ +

2πmkd

€

z = ±1 + r( )

2− ′ t 2

1− r( )2 − ′ t 2

Refractive index n Permittivity ε Permeability μ

Im n > 0Re n > 0

Im ε < 0 ??? Re ε > 0

Im μ > 0Re μ < 0

Energy Losses Q in a passive medium are always positive in spite of the fact that Im ε < 0

€

Q(ω) = ′ ′ ε | E |2 + ′ ′ μ | H |2

€

Q(ω) = 2ω | H |2 ′ ′ n (ω) ′ z (ω)

Q(ω) > 0, provided that Im n(ω) > 0 and Re z(ω) > 0

1d single-ring SRR: retrieved Re n() via cHEM inversionfor different length of the unit cell: 6x10x9 ... 6x10x14

TMM simulated 1d single-ring SRR:retrieved Re n() via cHEM inversion

for different resonance frequencies

Emulate small SRR gap: we fill the gap with dielectic, eg.eps=300Vacuum case as before

π/(Nz)

TMM simulated 1d single-ring SRR: retrieved eps() and mu() via cHEM inversion

for different resonance frequencies

Emulate small srr gap:we fill the gap with dielectic, eg. eps=300

Vacuum case as before

No negative Im εω and Im μω are observed !

TMM simulated 1d single-ring off-plane LHM: retrieved Re n() and Im n() via cHEM inversion

Re n()

Im n()

TMM simulated 1d single-ring off-plane LHM: retrieved ε() and μ() via cHEM

Intermediate summary:continuum homogeneous effective material (cHEM)

● cHEM inversion basically works, we find length-independent(!) effective material behavior

Re n(ω) seems to be cut-off at Brillouing zone. Discrepancy between n(ω) and z(ω): where is the resonance? Resonance/anti-resonance coupling. Negative imaginary parts in εω or μω Deformed resonances, i.e. unexpected shallow negative μω What is all this structure at higher frequencies?

but problems:

Model: Effective periodic material (PEM)

PEM analytic SRR model: retrieved n(), z() and eps(), mu() via cHEM

n() z()

eps() mu()

PEM analytic LHM model: retrieved n(ω), z() and eps(), mu() via cHEM

n() z()

eps() mu()

TMM simulated 1d single-ring SRR:retrieved (cHEM) + calculated (PEM-to-cHEM) n(), z()

Re n() Im n()

Re z() Im z()

TMM simulated 1d single-ring SRR:retrieved core+avrg eps(), mu() via lattice PEM inversion

eps,muSRR

eps,muLHM

3D Rings & Wires

cell = 2.5mm in X/Y/Z

ring side length = 2.2mm

wire width = 0.2mm

ring width = 0.2mm

opposed ring separation = 0.2mm

this structure is symmetric in 3D and also behaves almost same for different polarizationsee black and red curve below. different size for gap azimuthal is chosen, they are 0.3mm, 0.2mm and 0.1mm.

6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

kxEz gap=.3 kxEy gap=.3 kxEz gap=.2 kxEz gap=.1S21

Frequency (GHz)

Retrieved Z, n

9 10 11 12 13 14 150

2

4

6

8

10

kxEz gap=.3 kxEy gap=.3 kxEz gap=.2 kxEz gap=.1

Frequency (GHz)

Z

9 10 11 12 13 14 15-6

-5

-4

-3

-2

-1

0

Frequency (GHz)

n

Retrieved n and ε μ

6 8 10 12 14-6

-5

-4

-3

-2

-1

0

real n imag n

n

Frequency(GHz)6 8 10 12 14

-50

-40

-30

-20

-10

0

10

20

Frequency(GHz)

real ε realμ

ε μ

there are multiple negative index regions from retrieval code